varied solids loading while holding air mass flow constant, thus providing a measure of the effect of solids independent of the basic convective heat transfer. Second, he claims that “a glance” a t the authors’ Figure 5 , which is a plot of h/h, us. loading with linear coordinates, will give information on the size and variation of an exponent for loading. However, the authors’ Figure 6 plots Xusselt number us. loading on logarithmic coordinates, from which it is apparent that the exponent for loading is constant for the range of W,/W, from 2.0 (or slightly lower) to the highest ievels used; Farbar and hforley reported that the value of the exponent was 0.45. I reported (my paper) on the agreement of other data with this relationship, so will not repeat the discussion here. It can also be noted from the same Figure 6 that the value of h is substantially independent of solids content below a loading of 2.0 or slightly lower. This same effect is shown in the results of Wilkinson and Norman. It is obvious that entirely different relationships hold above and below the break point. Therefore, Sadek’s claim (his letter) on the basis of continuity that h for gas-solids mixtures must vary as the 0.8 power of the Reynolds number, because h for pure gas does, has no merit. The experimental data show that there is no continuity. Facts about Dense-Phase Runs in Danziger’s Paper. Sadek (letter) refers to the dense-phase runs a number of times; let us consider the facts. The two runs actually represent a plant-scale experiment made possible by a temporary revamp of piping; anyone with expertise in gas-solids flow would be able to predict that it would be impossible to obtain flow a t the conditions listed with the original setup as shown in my paper’s Figure 1. The revamp details, developed by engineers a t the refinery of the major oil company where the unit was located, are considered confidential. The hiickley
and Trilling (1949) apparatus would not permit similar operation. The mere fact that the densities were possibly comparable in some runs to those of the two dense-phase runs is no “proof” that the lliickley and Trilling runs were true transport. The short paragraph in my paper comparing h for fluidized beds with h for transport was a summary of a section of my original manuscript that went into some detail but that was excised as a result of your editorial decision that i t was only peripheral to the subject of my paper. Finding a continuity of correlation between beds and transport would have been a significant discovery; my statement that there was no continuity was a considered judgment based on my investigation. Sadek obviously did not make a n actual investigation before claiming I was wrong. Space limitations prevent the listing and correction of Sadek’s other misstatements. However, the items that have already been discussed in this letter provide a guide for evaluation of Sadek’s other statements about me and my paper. Literature Cited
Danziger, W. J., Ind. Eng. Chem., Process Des. Develop.,
2, 269
(1963).
Danziger, W. J., Ind. Eng. Chem., Process Des. Develop., 11, 634 (1972).
Farbar, L., Morley, M.W., Ind. Eng. Chem., 49, 1143 (1957). llickley, H. W., Trilling, C. A., Ind. Eng. Chem., 41, 1135 (1949). Sadek, S. E., Ind. Eng. Chem., Process Des. Develop., 1 1 , 133 (1972a).
Sadek, S. E., Ind. Eng. Chem., Process Des. Develop., 11, 635 (1972b).
Wilkinson, G. T., Norman, J. R., Trans. Inst. Chem. Eng.,
45,
T314 (1967).
Wilkinson, G. T., personal communication, 1972. W . J. Danziger
Bronx, S.Y .
Heat Transfer to Air-Solids Suspensions in Turbulent Flow. Modified Equations SIR: I n a recent article on the heat transfer to air-solids suspensions in turbulent flow (Sadek, 1972a), a relationship describing the increase in heat transfer coefficient as a function of a particle area parameter was developed based on data published in the open literature. T h a t relationship was derived by plotting the relative increase in heat transfer coefficient above that of particle-free air against a particle area parameter, nd2D,defined as 6 W,
nd2D =
-(E)(:)(:)
T
too
. 0
x
J e p s o n et 0 1 .
A
Forbar a n d Marley M i c k l e y a n d Trillinq
c
c
a
-
.‘y
10
L
u 0
(Y
and fitting visually a straight line through the data. The scatter in plotting the data was somewhat decreased by using the correct area-mean diameter of 30 p nhen reducing Farbar and Morley’s (1957) data instead of a diameter of 20 p as Sadek had previously used. When that correction, pointed out by Danziger (1972), was made. a modification of Sadek’s relationship became necessary. The modified relationship 15 as reported graphically by Sadek (1972b) and is shown here as the solid line on Figure 1. An approximate equation describing this relationship is (Ah/ho) = 0.5(nd2D)2/(1
-e8
1
u 0
‘y
-
5 0.1
+ nd2D)
T h a t relationship is shown in Figure 1. An attempt was made by Sadek (1972b) to compare his correlation to industrial data (Danziger, 1963). I n order to do SO, a number of assumptions were necessarily made. the mean particle diameter was assumed to be 10 p (ranging
P a r t i c l e Area P a r a m e t e r . n d 2 D
Figure 1 . Correlation between increase in heat transfer coefficient and particle area parameter Ind. Eng. Chern. Process Des. Develop., Vol. 12, No. 3, 1973
397
shown based on the corrected gas densities of 0.052 and 0.0139 lb/fta. A significant difference still exists between the industrial data and the relationship based on the experimental laboratory data. It is possible that the discrepancy is due to the agglomeration of the particles, to flow channelling and poor distribution through the equipment, or to fouling or other resistances which have not been accounted for. Nomenclature
particle diameter, ft pipe diameter, ft h = heat transfer coefficient, Btu/(hr ft2 O F ) n = average number of particles per unit gas volume in pipe,/ft3 V = velocity, ft/hr W = mass flow rate, lb/hr d
D
=
=
GREEKLETTERS A = increase beyond particle-free value p = density, lb/fts
SUBSCRIPTS g 0 p
=
= =
pertaining to gas in particle-free flow pertaining to particles
literature Cited
Particle A r e a Parameier. n d 2 D
Danziger, W. J., Ind. Eng. Chem., Process Des. Develop., 2, 269 Figure 2. Comparison between correlation and industrial scale data (Danziger, 1963). (Points calculated for a mean particle diameter of 40 p; lower and upper limits represent mean particle diameters of 50 and 30 p, respectively.)
between 30 and 50 w ) and the specific gravity of the particles was taken to be 2.46. The operating pressure was also assumed to be atmospheric. Contrary to Sadek’s last assumption, it was later reported (Danziger, 1973) that the operating pressures during these runs were significantly higher than atmospheric and that consequently the gas densities calculated were incorrect. These data have now been recalculated using the reported pressures of 11 psig for runs in the 1.5411. 0.d. tube units and 19 psig in the 1.8i5-in. 0.d. tube units, and are
(1963).
Danziger, W. J., Ind. Eng. Chem., Process Des. Develop., 11, 634 (1972).
Danziger, W. J., personal communication, 197’3. Farber, L., Depew, C. A., Ind. Eng. Chem., Fundam., 2, 130 (1963).
Farbar, L., Morley, M. W., Ind. Eng. Chem., 49, 1143 (1957’). Jepson, G., Poll, A., Smith, W., Trans. Inst. Chem. Eng., 41, 207 (1963).
Mickley, H. W., Trilling, C. A , , Ind. Eng. Chem., 41, 1135 (1949). Sadek, S. E., Ind. Eng. Chem., Process Des. Develop., 11, 133 (1972a).
Sadek, S. E., Ind. Eng. Chem., Process Des. Develop.,
11, 635
(197’2b).
Wilkinson, G. T., Norman, J. R., Trans. Znst. Chem. Eng.,
45,
T314 (1967).
S . E. Sadek Dynatech RID Company Cambridge, Massachusetts 0.2139
Point Efficiency in Binary Distillation with Unequal Molar Fluxes of Components SIR: I n a thorough investigation of mass transfer in binary mixtures with unequal molar heats of vaporization Todd and Van Winkle (1972) showed that the predicted values of Murphree efficiency fitted their experimental results well. The theoretical values of efficiency were based on rather complex equations of mass and heat transfer across the vapor-liquid interface. It is easy to show, however, that although the classical equation for molar flux which implies a n equimolar countertransfer gives incorrect results if the molar fluses are not equal, the number of mass transfer units and the point efficiency still can be calculated from the classical equations thanks to a successful cancellation of effects (Pohjola and Smigelschi, 1973). For the molar flus of the more volatile component the following holds (slightly simplifying the notation of Todd and Van Winkle) 398
Ind. Eng. Chem. Process Des. Develop., Vol. 12,
No. 3, 1973
where
r
=
X1/(N1
+ iV2)
(2)
By the heat balance across the vapor-liquid interface
r=
1
1-
(Xl/X,)
+ (q/lYlM
(3)
where q is the net heat flus between the interface and the bulk of a phase. When / r / is not extremely small good approximations for eq 1 are -lrl
= [r/(T
- Y)lkG(Yt
-
Y)
(4)
